“On the Merger of Two Companies,” Hans Gerber and Elias S. W. Shiu,July 2006

Abstract

Bankruptcy risk falls to pension plan participants if a plan sponsor fails when a defined benefit (DB) pension plan is underfunded. This article examines the incidence of that risk and how it changes when public policy provides a guarantee fund. Although government-based guarantee funds are in a unique position to provide pension protection, primarily because of the extent to which the risk of sponsor default is systematic in nature, a looming question is the extent to which such guarantees are exposed to moral hazard. The article focuses on that question using data from four Canadian provinces, including one (Ontario) that operates a guarantee fund for pensions. The findings show that plan assets per DB-plan participant increase with the earnings of workers and decrease with higher unemployment, and that level of assets also is moderated by the influence of taxes, with higher plan assets observed when and where tax rates are higher. Plans in Ontario had on average $20,035 less in asset value per participant, and Ontario plans covered by the guarantee fund had an average of $16,497 less per participant than other Canadian DB plans not backed by a guarantee fund. A separate model finds the presence of a guarantee fund to be one of a very small number of variables significant in explaining variability in the plans’ funded ratios. These empirical results are consistent with the existence of moral hazard.     

“Relative Importance of Risk Sources in Insurance Systems”, Edward W. Frees,April 1998

Abstract

This paper considers a Sparre Andersen collective risk model in which the distribution of the interclaim time is that of a sum of n independent exponential random variables; thus, the Erlang(n) model is a special case. The analysis is focused on the function φ(u), the expected discounted penalty at ruin, with u being the initial surplus. The penalty may depend on the deficit at ruin and possibly also on the surplus immediately before ruin. It is shown that the function φ(u) satisfies a certain integro-differential equation and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closed-form expression is obtained for the discounted joint probability density of the deficit at ruin and the surplus just before ruin, if the initial surplus is zero. For this formula and other results, the roots of Lundberg’s fundamental equation in the right half of the complex plane play a central role. Also, it is shown that φ(u) satisfies Li’s (2003) renewal equation. Under the assumption that the penalty depends only on the deficit at ruin and that the individual claim amount density is a combination of exponential densities, a closed-form expression for φ(u) is derived. In this context, known results of the Cauchy matrix are useful. Surprisingly, certain results are best expressed in terms of divided differences, a topic deleted from the actuarial examinations at the end of last century.